Final points from Christiaan

book/time_series/acf.md

@@ -12,7 +12,7 @@ Let us assume an arbitrary (discrete) stationary time series, $S=[S_1,S_2,...,S_
 The *formal* (or: theoretical) autocovariance is defined as
 
 $$
-Cov(S_t, S_{t-\tau}) =\mathbb{E}(S_tS_{t-\tau})-\mu^2
+Cov(S_t, S_{t+\tau}) =\mathbb{E}(S_tS_{t+\tau})-\mu^2
 =c_{\tau}
 $$
 
@@ -23,20 +23,20 @@ We have that $Cov(S_t, S_{t-\tau}) =Cov(S_t, S_{t+\tau})$.
 
 Show that the covariance can be written as: 
 
-$$Cov(S_t, S_{t-\tau}) = \mathbb{E}(S_tS_{t-\tau})-\mu^2
+$$Cov(S_t, S_{t+\tau}) = \mathbb{E}(S_tS_{t+\tau})-\mu^2
 =c_{\tau}$$ 
 
 
 ````{admonition} Solution
 :class: tip, dropdown
 
 $$
- Cov(S_t, S_{t-\tau})= \mathbb{E}[(S_t - \mathbb{E}(S_t))(S_{t-\tau} - \mathbb{E}(S_{t-\tau}))]\\
- = \mathbb{E}((S_t-\mu)(S_{t-\tau}-\mu))\\
- = \mathbb{E}(S_tS_{t-\tau} - \mu S_{t-\tau} - \mu S_t + \mu^2)\\
- = \mathbb{E}(S_tS_{t-\tau}) - \mu \mathbb{E}(S_{t-\tau}) - \mu \mathbb{E}(S_t) + \mu^2\\
-= \mathbb{E}(S_tS_{t-\tau}) - 2\mu^2 + \mu^2\\
-= \mathbb{E}(S_tS_{t-\tau}) - \mu^2\\
+ Cov(S_t, S_{t+\tau})= \mathbb{E}[(S_t - \mathbb{E}(S_t))(S_{t+\tau} - \mathbb{E}(S_{t+\tau}))]\\
+ = \mathbb{E}((S_t-\mu)(S_{t+\tau}-\mu))\\
+ = \mathbb{E}(S_tS_{t+\tau} - \mu S_{t+\tau} - \mu S_t + \mu^2)\\
+ = \mathbb{E}(S_tS_{t+\tau}) - \mu \mathbb{E}(S_{t+\tau}) - \mu \mathbb{E}(S_t) + \mu^2\\
+= \mathbb{E}(S_tS_{t+\tau}) - 2\mu^2 + \mu^2\\
+= \mathbb{E}(S_tS_{t+\tau}) - \mu^2\\
 $$
 ````
 :::
@@ -46,7 +46,6 @@ $$
 Prove that $Cov(S_t, S_{t-\tau}) =Cov(S_t, S_{t+\tau})$: 
 
 
-
 ````{admonition} Solution
 :class: tip, dropdown
 
@@ -77,7 +76,7 @@ $$ Cov(S_t, S_{t-\tau}) = Cov(S_t, S_{t+\tau})$$
 The *formal* autocorrelation is defined as
 
 $$
-r_{\tau} = \mathbb{E}(S_tS_{t-\tau})
+r_{\tau} = \mathbb{E}(S_tS_{t+\tau})
 $$
 
 ```{note}
@@ -265,42 +264,3 @@ $$
 
 ```
 :::
-
-<!-- ## Power spectral density
-
-The power spectral density (PSD) explains how the power (variance) of the signal is distributed over different frequencies. For instance, the PSD of a pure sine wave is flat *except* at its constituent frequency, where it will show a peak. Purely random noise has a flat power spectrum, indicating that all frequencies have an identical contribution to the variance of the signal! -->
-
-<!-- The power spectral density (PSD) explains how the power (variance) of the signal is distributed over different frequencies. For instance, the PSD of a pure sine wave is flat *except* at its constituent frequency, where it will show a peak. Purely random noise has a flat power spectrum, indicating that all frequencies have an identical contribution to the variance of the signal!
-
-### PSD vs ACF
-
-Knowledge on ACF, in time domain, is mathematically equivalent to knowledge on PSD, in the frequency domain, and vice-versa. And, from here, you might have a clue of where this is taking us... The PSD is the **discrete Fourier transform (DFT)** of the ACF.
-
-$$
-\begin{align*}
-\text{DFT}(\hat{c}_{\tau})&=\text{DFT}\left(\frac{1}{m}\sum_{i=1}^m s_i s_{i+\tau}\right)\\&=\frac{1}{m\Delta t}F_S(k) F_S(k)^*\\&=\frac{1}{m\Delta t}|F_S(k)|^2
-\end{align*}$$
-
-where the Fourier coefficients (see [DFT section](FFT)) are:
-
-$$F_S(k)  = \Delta t\sum_{i=1}^my_ie^{-j\frac{2\pi}{m}(k-1)(i-1)}$$
-
-```{note}
-In signal processing, it is common to write a sampled (discrete) signal as a small letter $s_i$ and the Fourier coefficients with capitals $S_k$. Since we also use capitals to indicate that $S$ is a random variable, we describe the DFT here for a realization $s_i$ of $S_i$, and use the notation $F_S(k)$ for the Fourier coefficients.
-```
-
-Conversely, the inverse discrete Fourier transform (IDFT) of the PSD is the ACF, so
-
-$$\text{IDFT}(F_{S}(k))=\hat{c}_{\tau}, \hspace{35px} \tau = 1,...,m \hspace{5px}\text{and}\hspace{5px} k = 1,...,m$$
-
-```{figure} ./figs/ACF_PSD.png
----
-height: 300px
-name: ACF_PSD
----
-Time series data, autocovariance and its power spectral density plots of white noise above and colored noise (not purely random) below.
-```
-
-The PSD explains how the power (variance) of the signal is distributed over different frequencies. The PSD of a pure sine wave is flat except at its constituent frequency.
-Purey random noise (i.e., white noise) has a flat power, indicating that all frequencies have identical contribution in making the variance of the signal. This is however not the case for time-correlated noise because different frequencies have different power values in making the total signal variability.
- -->

book/time_series/ar_exercise.ipynb

@@ -148,8 +148,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "AR(1) vs AR(2) test statistic: -2 Critical value: 3.841458820694124\n",
-      "Fail to reject AR(1)\n",
+      "AR(1) vs AR(2) test statistic: 65.46386398456401 Critical value: 3.841458820694124\n",
+      "Reject AR(1) in favor of AR(2)\n",
       "AR(2) vs AR(3) test statistic: 3.5318291727430835 Critical value: 3.841458820694124\n",
       "Fail to reject AR(2)\n"
      ]
@@ -186,7 +186,6 @@
     "dof = 1\n",
     "crit = chi2.ppf(0.95, dof)\n",
     "test_stat = n * np.log(rss1 / rss2)\n",
-    "test_stat = -2 \n",
     "print('AR(1) vs AR(2) test statistic:', test_stat, 'Critical value:', crit)\n",
     "\n",
     "if test_stat > crit:\n",
@@ -215,7 +214,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 6,
    "metadata": {},
    "outputs": [
     {

book/time_series/intro.md

@@ -9,5 +9,5 @@ Next, we will consider stationary time series, meaning that the statistical prop
 :width: 600px
 :align: center
 
-Recorded and expected global warming from 1960 to 2100 ([Huseien, Shah (2021)](https://www.mdpi.com/2071-1050/13/17/9720))
+Recorded and expected global warming from 1960 to 2100, from IPCC report ([Masson-Delmotte, et al. (20219)](https://www.researchgate.net/profile/Peter-Marcotullio/publication/330090901_Sustainable_development_poverty_eradication_and_reducing_inequalities_In_Global_warming_of_15C_An_IPCC_Special_Report/links/6386062b48124c2bc68128da/Sustainable-development-poverty-eradication-and-reducing-inequalities-In-Global-warming-of-15C-An-IPCC-Special-Report.pdf))
 ```

book/time_series/stationarity.md

@@ -3,7 +3,7 @@
 
 
 ```{admonition} Definition
-A stationary time series is a stochastic process whose statistical properties do not depend on the time at which it is observed.
+A stationary time series $S(t)$ is a stochastic process whose statistical properties do not depend on the time at which it is observed.
 ```
 
 This means that parameters such as *mean* and *(co)variance* should remain constant over time and not follow any trend, seasonality or irregularity.